Integrand size = 33, antiderivative size = 313 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-7 A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}} \]
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Time = 0.19 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 79, 43, 44, 65, 214} \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^4 (-3 a B e-7 A b e+10 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}}-\frac {e^3 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{128 b^2 (a+b x) (b d-a e)^4}+\frac {e^2 \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{192 b^2 (a+b x)^2 (b d-a e)^3}-\frac {e \sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{240 b^2 (a+b x)^3 (b d-a e)^2}-\frac {\sqrt {d+e x} (-3 a B e-7 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{3/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]
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Rule 27
Rule 43
Rule 44
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {(A+B x) \sqrt {d+e x}}{(a+b x)^6} \, dx \\ & = -\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-7 A b e-3 a B e) \int \frac {\sqrt {d+e x}}{(a+b x)^5} \, dx}{10 b (b d-a e)} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(e (10 b B d-7 A b e-3 a B e)) \int \frac {1}{(a+b x)^4 \sqrt {d+e x}} \, dx}{80 b^2 (b d-a e)} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^2 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{96 b^2 (b d-a e)^2} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^2 (b d-a e)^3} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^4 (10 b B d-7 A b e-3 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^2 (b d-a e)^4} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^3 (10 b B d-7 A b e-3 a B e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^2 (b d-a e)^4} \\ & = -\frac {(10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {e (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{240 b^2 (b d-a e)^2 (a+b x)^3}+\frac {e^2 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{192 b^2 (b d-a e)^3 (a+b x)^2}-\frac {e^3 (10 b B d-7 A b e-3 a B e) \sqrt {d+e x}}{128 b^2 (b d-a e)^4 (a+b x)}-\frac {(A b-a B) (d+e x)^{3/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-7 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{5/2} (b d-a e)^{9/2}} \\ \end{align*}
Time = 2.46 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {\sqrt {b} \sqrt {d+e x} \left (B \left (-45 a^5 e^4+30 a^4 b e^3 (4 d-7 e x)+2 a^3 b^2 e^2 \left (-218 d^2+409 d e x+192 e^2 x^2\right )-10 b^5 d x \left (48 d^3+8 d^2 e x-10 d e^2 x^2+15 e^3 x^3\right )+2 a^2 b^3 e \left (176 d^3-1178 d^2 e x-709 d e^2 x^2+105 e^3 x^3\right )+a b^4 \left (-96 d^4+1808 d^3 e x+484 d^2 e^2 x^2-730 d e^3 x^3+45 e^4 x^4\right )\right )+A b \left (-105 a^4 e^4+10 a^3 b e^3 (121 d+79 e x)+2 a^2 b^2 e^2 \left (-1052 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (744 d^3+128 d^2 e x-161 d e^2 x^2+245 e^3 x^3\right )+b^4 \left (-384 d^4-48 d^3 e x+56 d^2 e^2 x^2-70 d e^3 x^3+105 e^4 x^4\right )\right )\right )}{(b d-a e)^4 (a+b x)^5}+\frac {15 e^4 (-10 b B d+7 A b e+3 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{9/2}}}{1920 b^{5/2}} \]
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Time = 0.37 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.18
method | result | size |
pseudoelliptic | \(-\frac {7 \left (-\left (\left (A e -\frac {10 B d}{7}\right ) b +\frac {3 B a e}{7}\right ) \left (b x +a \right )^{5} e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+\sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, \left (\left (-A \,e^{4} x^{4}+\frac {2 \left (\frac {15 B x}{7}+A \right ) x^{3} d \,e^{3}}{3}-\frac {8 x^{2} d^{2} \left (\frac {25 B x}{14}+A \right ) e^{2}}{15}+\frac {16 x \left (\frac {5 B x}{3}+A \right ) d^{3} e}{35}+\frac {128 d^{4} \left (\frac {5 B x}{4}+A \right )}{35}\right ) b^{5}-\frac {496 \left (\frac {245 \left (\frac {9 B x}{98}+A \right ) x^{3} e^{4}}{744}-\frac {161 \left (\frac {365 B x}{161}+A \right ) x^{2} d \,e^{3}}{744}+\frac {16 x \,d^{2} \left (\frac {121 B x}{64}+A \right ) e^{2}}{93}+d^{3} \left (\frac {113 B x}{93}+A \right ) e -\frac {2 B \,d^{4}}{31}\right ) a \,b^{4}}{35}+\frac {2104 \left (-\frac {112 \left (\frac {15 B x}{64}+A \right ) x^{2} e^{3}}{263}+\frac {289 x d \left (\frac {709 B x}{289}+A \right ) e^{2}}{1052}+d^{2} \left (\frac {589 B x}{526}+A \right ) e -\frac {44 B \,d^{3}}{263}\right ) e \,a^{2} b^{3}}{105}-\frac {242 \left (\frac {79 \left (\frac {192 B x}{395}+A \right ) x \,e^{2}}{121}+d \left (\frac {409 B x}{605}+A \right ) e -\frac {218 B \,d^{2}}{605}\right ) e^{2} a^{3} b^{2}}{21}+\left (\left (2 B x +A \right ) e -\frac {8 B d}{7}\right ) e^{3} a^{4} b +\frac {3 B \,a^{5} e^{4}}{7}\right )\right )}{128 \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} b^{2} \left (a e -b d \right )^{4}}\) | \(368\) |
derivativedivides | \(2 e^{4} \left (\frac {\frac {\left (7 A b e +3 B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{256 e^{4} a^{4}-1024 b d \,e^{3} a^{3}+1536 b^{2} d^{2} e^{2} a^{2}-1024 b^{3} d^{3} e a +256 b^{4} d^{4}}+\frac {7 \left (7 A b e +3 B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {7}{2}}}{384 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 e^{2} a^{2}-60 a b d e +30 b^{2} d^{2}}+\frac {\left (79 A b e -21 B a e -58 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (7 A b e +3 B a e -10 B b d \right ) \sqrt {e x +d}}{256 b^{2}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{2} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(389\) |
default | \(2 e^{4} \left (\frac {\frac {\left (7 A b e +3 B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{256 e^{4} a^{4}-1024 b d \,e^{3} a^{3}+1536 b^{2} d^{2} e^{2} a^{2}-1024 b^{3} d^{3} e a +256 b^{4} d^{4}}+\frac {7 \left (7 A b e +3 B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {7}{2}}}{384 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 e^{2} a^{2}-60 a b d e +30 b^{2} d^{2}}+\frac {\left (79 A b e -21 B a e -58 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b \left (a e -b d \right )}-\frac {\left (7 A b e +3 B a e -10 B b d \right ) \sqrt {e x +d}}{256 b^{2}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {\left (7 A b e +3 B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{2} \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(389\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (281) = 562\).
Time = 0.44 (sec) , antiderivative size = 2580, normalized size of antiderivative = 8.24 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (281) = 562\).
Time = 0.29 (sec) , antiderivative size = 859, normalized size of antiderivative = 2.74 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {{\left (10 \, B b d e^{4} - 3 \, B a e^{5} - 7 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \sqrt {-b^{2} d + a b e}} - \frac {150 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 700 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 1280 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 580 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 150 \, \sqrt {e x + d} B b^{5} d^{5} e^{4} - 45 \, {\left (e x + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 105 \, {\left (e x + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 910 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 490 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 2944 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 1530 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 790 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 645 \, \sqrt {e x + d} B a b^{4} d^{4} e^{5} + 105 \, \sqrt {e x + d} A b^{5} d^{4} e^{5} - 210 \, {\left (e x + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 490 \, {\left (e x + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 2048 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 1792 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 1110 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 1080 \, \sqrt {e x + d} B a^{2} b^{3} d^{3} e^{6} - 420 \, \sqrt {e x + d} A a b^{4} d^{3} e^{6} - 384 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 50 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 2370 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 870 \, \sqrt {e x + d} B a^{3} b^{2} d^{2} e^{7} + 630 \, \sqrt {e x + d} A a^{2} b^{3} d^{2} e^{7} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 790 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 330 \, \sqrt {e x + d} B a^{4} b d e^{8} - 420 \, \sqrt {e x + d} A a^{3} b^{2} d e^{8} + 45 \, \sqrt {e x + d} B a^{5} e^{9} + 105 \, \sqrt {e x + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]
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Time = 11.24 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.80 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {7\,{\left (d+e\,x\right )}^{7/2}\,\left (7\,A\,b^2\,e^5-10\,B\,d\,b^2\,e^4+3\,B\,a\,b\,e^5\right )}{192\,{\left (a\,e-b\,d\right )}^3}-\frac {\sqrt {d+e\,x}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^2}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,{\left (a\,e-b\,d\right )}^2}+\frac {b^2\,{\left (d+e\,x\right )}^{9/2}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^4}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (21\,B\,a\,e^5-79\,A\,b\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (7\,A\,b\,e^5+3\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (7\,A\,b\,e+3\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{5/2}\,{\left (a\,e-b\,d\right )}^{9/2}} \]
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